Multiple Regression Analysis & Modelling - Coursework Example

Multiple regression Case Name: Number: Course: Lecturer: Date: Outline Title page Table of contents Statistical results Correlation coefficient the geometrical variables Simple Regression on Speed and Bendiness Regression equation t ratio for the slope coefficient : r2 value and the standard deviation of the errors: Analysis of Variance Simple Regression on Speed and hilliness Regression equation: t ratio for the slope coefficient : r2 value and the standard deviation of the errors Analysis of Variance Simple Regression on Speed and visibility t ratio for the slope coefficient : r2 value and the standard deviation of the errors: Analysis of Variance Multiple linear regression equations Analysis and interpretation of the results Works Cited Appendix Name Professor Subject Date Statistical results Correlation coefficient Correlation coefficient shows the relatedness of the variables in consideration. When the correction coefficient is negative, it means that the relationship is negative and if it is positive it means the relationship between the variables is positive. If the correlation coefficient is equal to one, the variables move in the same direction in a perfect linear relationship. If the correlation coefficient is greater than zero but less than one, there is no direct relationship although they move to the same direction. If the correlation coefficient is equal to zero means the variables have no direct relationship. When it is negative one, the variables have a direct negative relationship. Appendix 1 relating to plots shows correlation coefficient of speed against each variable is as follows bellow.   Mean free-flow speed (kph) heaviness bendiness Visibility carriargeway width Hardstrip width Verge width Number of Junctions Hilliness Mean free-flow speed (kph) 1 heaviness 0.07002 1 bendiness -0.7763 0.04468 1 Visibility 0.59998 0.23157 -0.54647 1 carriargeway width 0.50426 0.3176 -0.3965 0.59136 1 Hardstrip width 0.45777 0.326 -0.3195 0.35782 0.35143 1 Verge width 0.31063 0.2781 -0.3709 0.18884 0.30412 0.44419 1 Number of Junctions -0.0552 0.0259 -0.0374 -0.0984 -0.1392 -0.0267 -0.0633 1 Hilliness -0.26919 -0.12011 0.209256 -0.34885 -0.02742 -0.42184 -0.07969 -0.14061 1 From the above table, three variables are seen to have negative variables while five have a positive influence. It can be noted that variable Bendiness measure has a great negative influence on speed followed by hilliness and lastly number of junction’s. On the positive influence there is visibility followed by carriage way width, hard strip width, verge width and lastly proportion of heavy. It means that when there is an increase in visibility, carriage width, and hard strip width there is an increase in speed while when there is an increase in the number of junctions, Hills, and bendiness, the speed is reduced. This is according to expectation from the results. However it should be noted that the correlations of proportion of heavy vehicles and number of junctions have their values of correlation near zero meaning that their influence is minimal. From the geometrical variables carriage way width has the highest correlation coefficient of 0.504 while from explanatory variable visibility has the highest correlation coefficient. Simple Regression on Speed and Bendiness Regression equation: Regression is a method of estimating the linearity of two variables that is it determines the relationship between independent and dependent variable. This leads to formation of an equation of y=b+ax+e where e –is an-correlated error variables with a mean of zero and a common variance, a is the gradient of the linear line and b is the intercept. From the excel output on regression output from excel file for speed and bendiness shown below we note that the equation formed is y=84.45-0.116x where x is degrees turned through per kilometre and y is the mean flow speed. The equation is reasonable because it shows that an increase in bendiness reduces the mean free flow speed. It means when there is change in one unit of kilometre per hour on a bend there is a negative change of 0.116. The intercept is the value of dependent variable (bendiness) in the absence of independent variable. It means that without bendiness degrees, the mean free flow speed will be 84.45kph. t ratio for the slope coefficient : Another important factor to be looked at from the excel output is t-ratio is -7.79 while the p- value for the slope coefficient (1.5x10-9)% which is not significant from zero therefore we can conclude that the true regression parameters are significant in determining the future predictions. r2 value and the standard deviation of the errors: The standard error of the estimate is similar to the standard deviation, but the mean is not used. The standard error of the estimate is the square root of the unexplained variation- that is, the variation due to the difference of the observed values and the expected values. So the closer the observed values are to the predicted values, the smaller the standard error of the estimate. The coefficient of determination is a measure of the variation of the dependent variable that is explained by the regression line and the independent variable. The symbol for the coefficient of determination is r2.Looking at the r2 and standard deviation of errors we note that r2 is 0.6 and a standard deviation error is 6.02. We can state that it is significant because r2 suggest that only 62% of the data is represented in the linear regression at 95% level of significance with an error of 6. It is interesting to note that out of 42 observations only 27 observations are included in the regression analysis while the rest are outliers. Analysis of Variance – in the output, the residual sum of squares (RSS) is called "Error SS" is 1449.765 and the original sum of squares (TSS) is denoted by "Total SS" is 3647.989 while the regression SS is 2198.133. From the regression Analysis of Variance section TSS is 3647.898 while RSS is 1449.765. The coefficient of determination (r2) is arrived as =0.602. this confirms the answer obtained from summary output of excel The standard error will be determined as where RSS is the residual sum of squares and n is number of observations. From the excel output we know that RSS is 1449.765 and n is 42 s= = = 6.02 If those observations which have large influence on the regression line the regression equation will change including the relation of the variables. Simple Regression on Speed and hilliness Regression equation: From the excel output on regression output from excel file for speed and bendiness shown below we note that the equation formed is y=80.19-0.203x where x is metres of fall or rise per kilometre and y is the mean free-flow speed. The equation is reasonable because it shows that an increase in metres of rise reduces the mean free-flow speed. It means when there is change in one unit of kilometre per hour when there is a hill there is a negative change of 0.203. The intercept is the value of dependent variable (metres of fall or rise per kilometre) in the absence of independent variable. It means that without metres of fall or rise per kilometre, the mean free flow speed will be 80.19kph. t ratio for the slope coefficient : Another important factor to be looked at from the excel output is t-ratio is -1.77 while the p- value for the slope coefficient 8.47% which is significant from zero therefore we can conclude that the true regression parameters are not significant in determining the future predictions. r2 value and the standard deviation of the errors: Looking at the r2 and standard deviation of errors from the excel output, it can be noted that r2 is 0.07 and a standard deviation error is 9.2.one can state that it is insignificant because r2 suggest that only 7% of the data is represented in the linear regression at 95% level of significance with an error of 9.2. It is interesting to note that out of 42 observations only 3 observations are included in the regression analysis while the rest are outliers. Analysis of Variance – in the output, the residual sum of squares (RSS) is called "Error SS" is 3383.565 and the original sum of squares (TSS) is denoted by "Total SS" is 3647.898 while the regression SS is 264.3336. The coefficient of determination (r2) is arrived as =0.072 this confirms the answer obtained from summary output of excel The standard error will be determined as where RSS is the residual sum of squares and n is number of observations. From the excel output we know that RSS is 3383.565and n is 42 s= = = 9.2 If those observations which have large influence on the regression line the regression equation will change including the relation of the variables. Simple Regression on Speed and visibility The excel file shows that the intercept is 64.42 and the slope is 0.067. this gives a regression equation of y=64.42+0.067x where x is visibility and y is the mean free-flow speed. The equation is reasonable because it shows that an increase in visibility leads to increase in the mean free flow speed. The intercept is the value of mean free-flow speed in the absence of visibility. It means that without visibility, the mean free flow speed will be 64.42kph. t ratio for the slope coefficient : The t value as shown from the excel output is 4.74 with a p value 0.000027 which 0.0027% this is not significant from zero therefore we can conclude that the true regression parameters are significant in determining the future predictions. r2 value and the standard deviation of the errors: The other variables that will be looked at is coefficient of determination, r2 and standard deviation of errors which is 0.36 and 7.63 respectively. The coefficient of determination indicates that only 36% of the observations are included in the regression and the rest are outliers. Analysis of Variance – in the output, the residual sum of squares (RSS) is called "Error SS" is 2334.735 and the original sum of squares (TSS) is denoted by "Total SS" is 3647.989 while the regression SS is 1313.164. The coefficient of determination (r2) is arrived as =0.36. This confirms the answer obtained from summary output of excel The standard error will be determined as where RSS is the residual sum of squares and n is number of observations. From the excel output we know that RSS is 1449.765 and n is 42 s= = = 7.64 If those observations which have large influence on the regression line the regression equation will change including the relation of the variables. Multiple linear regression equations- appendix 3 From the multiple/stepwise regression equation the noted r2 and t-ratio values for the introduced is as follows: model Variables r2 t ratio new variable Comment 1 1 0.005 0.44 Only heaviness 2 2 0.594 -7.84 Bendiness ought to be included as it has increased the number of observations within the regression line 3 3 0.65 1.9 Visibility variable has improved the value of r2 thus useful to multiple regression equation. 4 4 0.66 1.27 Carriageway width is worth increasing to the equation as r2 increases. 5 5 0.69 1.74 Hardstrip width is useful to the equation 6 6 0.69 -0.58 Verge width has little influence on the number observations that will be within regression equation. But it goo for inclusion 7 7 0.694 1.26 Number of junctions has little influence thus its inclusion is indifference 8 8 0.695 0.088 Hilliness has little influence thus its inclusion is indifference The stepwise regression indicates the coefficient of determination R-squared is increasing when a variable is introduced which suggests that the stepwise regression performed between variables to predict the total variations observed in the values improves. This could be due to the seasonal trend in the values which is not addressed in this analysis. Also, due to the certain values that could be considered as outliers observed within the data. Analysis and interpretation of the results The analysis shows significant positive correlation between Mean free-flow speed with Proportion of heavy vehicles, Visibility, Carriageway width, Hardstrip width and Verge width confirming what one would intuitively expect. The positive correlation between Mean free-flow speed and increase in the Carriageway width confirms the findings of earlier research that reports similar results 5 years ago. Significantly, Proportion of heavy vehicles do not appear to high affect speed as they show a near zero positive correlation. The only interpretation is that this factor affects the Mean free-flow speed. Further, while Carriageway width has declined, the Mean free-flow speed has risen emphasizing the fact that Carriageway width on a countrywide basis has high effect on Mean free-flow speed. Similarly, the strong negative correlation between increase in Bendiness measure, Number of junctions per km and Hilliness measure and Mean free-flow speed appear follow popular misconceptions. Regression Analysis Based on this the following equation represents the stepwise empirical models for the analysis: Model Regression equation 1 Y (speed) = 75.77 + 9.2heaviness 2 Y (speed) = 82.67+ 13.79 heaviness – 0.1172bendiness 3 Y (speed) = 77.55+ 5.89heaviness – 0.098bendiness+ 0.027 visibility 4 Y (speed) = 69.84+ 1.38heaviness – 0.093bendiness+ 0.018 visibility+ 1.23 carriageway width 5 Y (speed) = 70.31- 5.4heaviness – 0.087bendiness+ 0.016 visibility+ 1.07 carriageway width + 4.78Hardstrip width 6 Y (speed) = 71.11 – 3.66heaviness – 0.091bendiness+ 0.014 visibility+ 1.14 carriageway width + 5.2Hardstrip width – 0.418verge width 7 Y (speed) = 72.25 – 2.66heaviness – 0.093bendiness+ 0.014 visibility+ 1.08 carriageway width + 5.28Hardstrip width – 0.45verge width – 0.59 number of Junctions 8 Y (speed) = 72.4 – 2.69heaviness – 0.092bendiness+ 0.012 visibility+ 1.16 carriageway width + 4.9Hardstrip width – 0.44verge width – 0.65 number of Junctions - 0.024 Hilliness The model does not incorporate a noise term based on the assumption that this is zero on average in a good regression fit. Since the regression model developed is an estimation using the sample data, we need some extensions in the analyses to check that the sample model developed is enough to represent the population. We have to conduct tests for correlation coefficient, regression parameters and regression model. The null hypothesis claims for zero population correlation coefficient. The result of the traditional F-Test is a significantly high 9.4 helps reject the null hypothesis that none of the selected independent variables has power to explain the variations in the dependent variables. Deploying the F-test (p-value) delivers a similar result, in that because of the significant difference (5% > 0.00012%) between the 5% hypothesis and the probable F-statistic the model has significant power to explain the changes in mean free-flow rate. In addition, the R-Square value of 69.5% shows that there exists a good fit between the independent and the dependent variables. Hence we can conclude that the population linear relationship is significant. Using the independent variables separately, the null hypothesis that the coefficients are =0 i.e. they have no effect whatsoever on the dependent variable is not borne out by the results as the P-values in the case of Proportion of heavy vehicles, Visibility, Carriageway width, Hard strip width, Verge width, Number of junctions per km and Hilliness measure are significantly higher than the threshold value of 5%. This indicates that these factors significantly influence mean-free flow speed. On the other hand, bendiness has little significance also borne out by the correlation analysis. Because mean-free flow speed are very small numbers themselves, the coefficients appear as small numbers also but the effect is important if we look at these as percent changes in mean-free flow speed. Works Cited Taha, Hamdy. Operations Research - An Introduction. New Delhi: Pearson Education Pte Ltd, 2005. Print Appendices Appendix 1: Plots Appendix 2: visibility SUMMARY OUTPUT Regression Statistics Multiple R 0.599982 R Square 0.359978 Adjusted R Square 0.343978 Standard Error 7.639919 Observations 42 ANOVA   df SS MS F Significance F Regression 1 1313.164 1313.164 22.49786 2.67E-05 Residual 40 2334.735 58.36837 Total 41 3647.898         Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 64.42415 2.900376 22.21234 4.12E-24 58.56227 70.28603 58.56227 70.28603 visibility 0.067293 0.014187 4.743191 2.67E-05 0.03862 0.095967 0.03862 0.095967 hilliness SUMMARY OUTPUT Regression Statistics Multiple R 0.269187 R Square 0.072462 Adjusted R Square 0.049273 Standard Error 9.197234 Observations 42 ANOVA   df SS MS F Significance F Regression 1 264.3336 264.3336 3.124913 0.084732 Residual 40 3383.565 84.58912 Total 41 3647.898         Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 80.1933 2.300077 34.86548 1.48E-31 75.54467 84.84193 75.54467 84.84193 hilliness -0.20343 0.115081 -1.76774 0.084732 -0.43602 0.029154 -0.43602 0.029154 bendiness SUMMARY OUTPUT Regression Statistics Multiple R 0.776257 R Square 0.602575 Adjusted R Square 0.59264 Standard Error 6.02031 Observations 42 ANOVA   df SS MS F Significance F Regression 1 2198.133 2198.133 60.64795807 1.54E-09 Residual 40 1449.765 36.24414 Total 41 3647.898         Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 84.45057 1.334104 63.30132 9.96926E-42 81.75425 87.1469 81.75425 87.1469 bendiness -0.11647 0.014956 -7.78768 1.53712E-09 -0.1467 -0.08625 -0.1467 -0.08625 Appendix 3: SUMMARY OUTPUT Regression Statistics Multiple R 0.070015 R Square 0.004902 Adjusted R Square -0.01998 Standard Error 9.5263 Observations 42 ANOVA   df SS MS F Significance F Regression 1 17.88235 17.88235 0.19705 0.65950489 Residual 40 3630.016 90.7504 Total 41 3647.898         Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 75.77491 3.114085 24.33296 1.37E-25 69.4811143 82.06871 69.48111 82.06871 heaviness 9.20072 20.72687 0.443903 0.659505 -32.689847 51.09129 -32.6898 51.09129 SUMMARY OUTPUT Regression Statistics Multiple R 0.7833 R Square 0.613558 Adjusted R Square 0.593741 Standard Error 6.012168 Observations 42 ANOVA   df SS MS F Significance F Regression 2 2238.198 1119.099 30.96037 8.8744E-09 Residual 39 1409.701 36.14617 Total 41 3647.898         Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 82.66966 2.153242 38.39311 1.34E-32 78.314315 87.025 78.31432 87.025 heaviness 13.78559 13.09406 1.052812 0.298906 -12.69965 40.27084 -12.6996 40.27084 bendiness -0.11718 0.014951 -7.83748 1.56E-09 -0.1474161 -0.08693 -0.14742 -0.08693 SUMMARY OUTPUT Regression Statistics Multiple R 0.805281 R Square 0.648477 Adjusted R Square 0.620726 Standard Error 5.809065 Observations 42 ANOVA   df SS MS F Significance F Regression 3 2365.579 788.5264 23.36704 9.6625E-09 Residual 38 1282.319 33.74524 Total 41 3647.898         Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 77.55053 3.357186 23.09986 5.68E-24 70.7542647 84.3468 70.75426 84.3468 heaviness 5.886797 13.28888 0.442987 0.660287 -21.015131 32.78873 -21.0151 32.78873 bendiness -0.09756 0.017624 -5.53575 2.47E-06 -0.1332383 -0.06188 -0.13324 -0.06188 Visibility 0.026284 0.013529 1.942884 0.059466 -0.0011027 0.053671 -0.0011 0.053671 SUMMARY OUTPUT Regression Statistics Multiple R 0.814317 R Square 0.663113 Adjusted R Square 0.626693 Standard Error 5.763188 Observations 42 ANOVA   df SS MS F Significance F Regression 4 2418.968 604.742 18.20726 2.4058E-08 Residual 37 1228.93 33.21433 Total 41 3647.898         Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 69.83941 6.934381 10.07147 3.78E-12 55.7890202 83.8898 55.78902 83.8898 heaviness 1.377758 13.65521 0.100896 0.920178 -26.290316 29.04583 -26.2903 29.04583 bendiness -0.09385 0.017728 -5.29391 5.67E-06 -0.12977 -0.05793 -0.12977 -0.05793 Visibility 0.018324 0.014818 1.236613 0.224023 -0.0116998 0.048347 -0.0117 0.048347 carriargeway width 1.230603 0.970634 1.267833 0.212778 -0.7360897 3.197295 -0.73609 3.197295 SUMMARY OUTPUT Regression Statistics Multiple R 0.830257 R Square 0.689327 Adjusted R Square 0.646178 Standard Error 5.610765 Observations 42 ANOVA   df SS MS F Significance F Regression 5 2514.594 502.9187 15.97547 2.7376E-08 Residual 36 1133.305 31.48069 Total 41 3647.898         Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 70.31519 6.756501 10.40704 2.12E-12 56.6123687 84.018 56.61237 84.018 heaviness -5.4002 13.85121 -0.38987 0.698928 -33.491753 22.69136 -33.4918 22.69136 bendiness -0.08736 0.017656 -4.94808 1.76E-05 -0.1231703 -0.05155 -0.12317 -0.05155 Visibility 0.016234 0.014476 1.121482 0.269508 -0.0131237 0.045592 -0.01312 0.045592 carriargeway width 1.067432 0.94959 1.124098 0.268411 -0.8584257 2.99329 -0.85843 2.99329 Hardstrip width 4.755019 2.72827 1.74287 0.089893 -0.7781678 10.28821 -0.77817 10.28821 SUMMARY OUTPUT Regression Statistics Multiple R 0.832049 R Square 0.692305 Adjusted R Square 0.639558 Standard Error 5.663011 Observations 42 ANOVA   df SS MS F Significance F Regression 6 2525.459 420.9098 13.12485 9.9933E-08 Residual 35 1122.439 32.0697 Total 41 3647.898         Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 71.10585 6.953386 10.22608 4.71E-12 56.9897249 85.22197 56.98972 85.22197 heaviness -3.55509 14.33506 -0.248 0.805584 -32.656813 25.54663 -32.6568 25.54663 bendiness -0.09101 0.018889 -4.81811 2.78E-05 -0.1293541 -0.05266 -0.12935 -0.05266 Visibility 0.01441 0.014943 0.964382 0.341471 -0.0159247 0.044745 -0.01592 0.044745 carriargeway width 1.135895 0.965623 1.176334 0.2474 -0.824423 3.096213 -0.82442 3.096213 Hardstrip width 5.263397 2.888866 1.82196 0.077016 -0.6013123 11.12811 -0.60131 11.12811 Verge width -0.4176 0.717439 -0.58207 0.564251 -1.8740776 1.038879 -1.87408 1.038879 SUMMARY OUTPUT Regression Statistics Multiple R 0.833236 R Square 0.694282 Adjusted R Square 0.63134 Standard Error 5.727205 Observations 42 ANOVA   df SS MS F Significance F Regression 7 2532.669 361.8098 11.03049 3.4771E-07 Residual 34 1115.23 32.80088 Total 41 3647.898         Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 72.24848 7.442577 9.707455 2.48E-11 57.1233453 87.37362 57.12335 87.37362 heaviness -2.66146 14.62233 -0.18201 0.856653 -32.3776 27.05468 -32.3776 27.05468 bendiness -0.09256 0.019387 -4.77427 3.36E-05 -0.1319571 -0.05316 -0.13196 -0.05316 Visibility 0.013669 0.015194 0.899622 0.374649 -0.0172095 0.044548 -0.01721 0.044548 carriargeway width 1.078633 0.984177 1.095975 0.280792 -0.9214546 3.078721 -0.92145 3.078721 Hardstrip width 5.278448 2.921789 1.806581 0.079686 -0.6593421 11.21624 -0.65934 11.21624 Verge width -0.45175 0.72922 -0.6195 0.539715 -1.9337063 1.030199 -1.93371 1.030199 Number of Junctions -0.59045 1.259414 -0.46883 0.642186 -3.149885 1.96899 -3.14989 1.96899 SUMMARY OUTPUT Regression Statistics Multiple R 0.833647 R Square 0.694968 Adjusted R Square 0.621021 Standard Error 5.806804 Observations 42 ANOVA   df SS MS F Significance F Regression 8 2535.172 316.8965 9.398167 0.0000012 Residual 33 1112.726 33.71897 Total 41 3647.898         Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 72.39789 7.565911 9.568959 0.00000 57.0049281 87.79085 57.00493 87.79085 heaviness -2.6943 14.82604 -0.18173 0.856908 -32.858112 27.46951 -32.8581 27.46951 bendiness -0.09256 0.019656 -4.70873 4.34E-05 -0.1325476 -0.05257 -0.13255 -0.05257 Visibility 0.01217 0.016358 0.743959 0.462166 -0.0211116 0.045452 -0.02111 0.045452 carriargeway width 1.16098 1.042612 1.11353 0.273526 -0.9602298 3.28219 -0.96023 3.28219 Hardstrip width 4.930325 3.226134 1.528246 0.135981 -1.6332934 11.49394 -1.63329 11.49394 Verge width -0.43526 0.741829 -0.58674 0.561374 -1.9445199 1.074004 -1.94452 1.074004 Number of Junctions -0.6492 1.294992 -0.50131 0.619478 -3.2838783 1.985484 -3.28388 1.985484 Hilliness -0.02419 0.088789 -0.27249 0.786942 -0.2048369 0.156448 -0.20484 0.156448 Read More